Question

the hypotenuse of a right triangle is 11 inches. if one leg is 5 inches, find the degree measure of each angle. the angle opposite the 5 - inch leg is (square^{circ}). (do not round until the final answer. then round to one decimal place as needed.) the third angle of the right triangle is (square^{circ}). (do not round until the final answer. then round to one decimal place as needed.)

Explanation:

Step1: Find the angle opposite the 5 - inch leg

We know that in a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Let \(\theta\) be the angle opposite the 5 - inch leg. Given \(\text{opposite} = 5\) and \(\text{hypotenuse}=11\).
So, \(\sin\theta=\frac{5}{11}\).
Using the inverse sine function \(\theta=\sin^{- 1}(\frac{5}{11})\).
\(\theta=\sin^{-1}(0.454545\cdots)\approx27.0^{\circ}\) (using a calculator, \(\sin^{-1}(x)\) where \(x = \frac{5}{11}\)).

Step2: Find the third angle

In a right - triangle, the sum of the angles is \(180^{\circ}\), and one angle is \(90^{\circ}\). Let the third angle be \(\alpha\).
We know that \(\alpha=90^{\circ}-\theta\).
Since \(\theta\approx27.0^{\circ}\), then \(\alpha = 90^{\circ}-27.0^{\circ}=63.0^{\circ}\)

Answer:

The angle opposite the 5 - inch leg is \(27.0^{\circ}\), and the third angle of the right - triangle is \(63.0^{\circ}\)